# Damped Oscillator

The Damped Oscillator is an example that is commonly used in structural reliability to demonstrate the robustness of the proposed approach for estimating low failure probabilities and their gradients.

## Description

Figure 1 shows the considered two-degree-of-freedom primary-secondary system which is characterized by the masses m_p and m_s, spring stiffnesses k_p and k_s and damping ratios \zeta_p and \zeta_s. The subscripts p and s refer to the primary and secondary oscillators. Finally, the excitation, given by a white noise base acceleration, is represented through S(t).

Figure 1: A two-d.o.f. damped oscillator under a white-noise base acceleration, adapted from Dubourg (2011).

Assuming that the reliability of the entire system can be reduced to the capacity of the secondary spring, one finds a performance function which is defined as follows:

g(x) = F_s - k_s \max_{t \in [0, \tau]} |x_s(t)|

where

• F_s: Force capacity of the secondary spring
• \tau: Duration of the excitation
• x_s: Relative displacement

In the sequel, the maximum is substituted by its mean amplitude since the uncertainty in the peak response is usually small. Hence, the performance function can be rewritten:

g(x) = F_s - p \cdot k_s \sqrt{E_S [x_s^2]}

where

• p: So-called peak factor, set equal to 3

The mean-square relative displacement of the secondary spring under a white noise base acceleration S can be calculated using the analytic expression shown below:

E_S [x_s^2] = \frac{\pi S_0}{4\zeta_s \omega_s^3} \cdot \frac{\zeta_a \zeta_s}{\zeta_p \zeta_s (4\zeta_a^2 + \theta^2) + \gamma \zeta_a^2} \cdot \frac{(\zeta_p \omega_p^3 + \zeta_s \omega_s^3) \omega_p}{4\zeta_a \omega_a^4}

where

• \omega_p, \omega_s: Natural frequencies, given as
• \omega_p = \sqrt{\frac{k_p}{m_p}}, \omega_s = \sqrt{\frac{k_s}{m_s}}
• S_0: Intensity of the white noise
• \gamma, \omega_a, \zeta_a, \theta: Further abbreviations and ratios, determined through
• \gamma = \frac{m_s}{m_p}
• \omega_a = \frac{\omega_p + \omega_s}{2}
• \zeta_a = \frac{\zeta_p + \zeta_s}{2}
• \theta = \frac{\omega_p - \omega_s}{\omega_a}

## Inputs

The Damped Oscillator problem contains eight independent, lognormal distributed random variables X = \{m_p, m_s, k_p, k_s, \zeta_p, \zeta_s, S_0, F_s\}:

No Variable Description Distribution Statistics
1 m_p Primary mass Lognormal \mu_{m_p} = 1.5,
\text{CV}_{m_p} = 10\%
2 m_s Secondary mass Lognormal \mu_{m_s} = 0.01,
\text{CV}_{m_s} = 10\%
3 k_p Primary spring stiffness Lognormal \mu_{k_p} = 1,
\text{CV}_{k_p} = 20\%
4 k_s Secondary spring stiffness Lognormal \mu_{k_s} = 0.01,
\text{CV}_{k_s} = 20\%
5 \zeta_p Primary damping ratio Lognormal \mu_{\zeta_p} = 0.05,
\text{CV}_{\zeta_p} = 40\%
6 \zeta_s Secondary damping ratio Lognormal \mu_{\zeta_s} = 0.02,
\text{CV}_{\zeta_s} = 50\%
7 S_0 Intensity of the white noise Lognormal \mu_{S_0} = 100,
\text{CV}_{S_0} = 10\%
8 F_s Force capacity of the secondary spring Lognormal \mu_{F_s} = \{15, 21.5, 27.5\},
\text{CV}_{F_s} = 10\%

### Constants

The default value of the only constant, the peak factor, is determined as follows:

Constant Value
p 3

## Reference results

The following table lists reference results of the probability of failure P_f for different values of the force capacity of the secondary spring F_s using subset simulation (Dubourg, 2011).

\mu_{F_s} 15 21.5 27.5
Number of samples N 300{,}000 500{,}000 700{,}000
Probability of failure P_f 4.63 \times 10^{-3} 4.75 \times 10^{-5} 3.47 \times 10^{-7}
Coefficient of variation CV <3\% <4\% <5\%

## Resources

The vectorized implementation of the damped oscillator in MATLAB, as well as the script file with the model and probabilistic inputs definitions for the function in UQLab, can be downloaded below:

uq_dampedOscillator.zip (2.9 KB)

The contents of the file are:

Filename Description
uq_DampedOscillator.m vectorized implementation of the damped oscillator performance function in MATLAB
uq_Example_dampedOscillator.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## Open-access repository

The dataset used in this benchmark study is titled “Benchmark case datasets - Damped oscillator” and is authored by Adéla Hlobilová, Stefano Marelli, and Bruno Sudret. It was published in 2024 and is available on Zenodo. The dataset can be accessed directly via the following DOI link: 10.5281/zenodo.12700167.

The experimental designs include datasets with 400, 800, 1200, 1600, and 2000 samples, each generated using optimized Latin Hypercube Sampling (LHS) with 1,000 iterations to improve the maximin criterion. Each dataset is replicated 20 times. The validation set contains 100,000 samples generated by Monte Carlo simulation. Each dataset contains samples and responses of the computational model.

For citation purposes, please use the following format:
Hlobilová, A., Marelli, S., and Sudret, B. (2024). Benchmark case datasets - Damped oscillator. Zenodo. https://doi.org/10.5281/zenodo.12700167.

This project was supported by the Open Research Data Program of the ETH Board under Grant number EPFL SCR0902285.

## References

• Igusa, T., A. Der Kiureghian “Dynamic characterization of two degree-of-freedom equipment-structure systems,” Journal of Engineering Mechanics, vol. 111, issue 1, pp. 1–19, 1985. DOI:10.1061/(ASCE)0733-9399(1985)111:1(1)
• Der Kiureghian, A., De Stefano, M. “An efficient algorithm for second-order reliability analysis,” Technical Report UCB/SEMM-90/20, Department of Civil and Environmental Engineering, University of California, Berkeley, 1990.
• Dubourg, V. Adaptive surrogate models for reliability analysis and reliability-based design optimization. Doctoral thesis, Blaise Pascal University – Clermont II, 2011. PDF file
• Lüthen, N., Marelli, S., Sudret, B. “Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark,” SIAM/ASA Journal on Uncertainty Quantification, vol. 9, issue 2, pp. 593–649, 2021. DOI:10.1137/20M1315774